= Solution
{c}
The <uniqueness theorem for moment-generating functions> states: if two real <random variables> $V,W$ have finite <moment-generating functions> on a common open <interval> $(-\delta,\delta)$ with $\delta>0$, and
$$
M_V(t)=\mathbb E e^{tV}=\mathbb E e^{tW}=M_W(t)\qquad(|t|<\delta),
$$
then $\boxed{V\text{ and }W\text{ have the same probability distribution}.}$ Finiteness throughout a neighborhood of zero is a hypothesis; the identity $M_V(0)=M_W(0)=1$ alone carries no information.
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